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Extremal structures of multiphase heat conducting composites
In,
Pergamon 0020-7683(95)00176-X
EXTREMAL
STRUCTURES CONDUCTING
J. Solidr Strucrures
Vol. 33, No. 18, pp. 2609-2623, 1996 0 1996 Elsevier Science Ltd Copyright Pnnted-in &at Britain. All rights reserved 002&7683/96 $15.00 + .oO
OF MULTIPHASE COMPOSITES
HEAT
A. V. CHERKAEV University
of Utah, Department
of Mathematics,
Salt Lake City, UT 84112, U.S.A.
and L. V. GIBIANSKY Princetown
University,
Department of Civil Engineering Princeton, NJ 08544, U.S.A.
and Operations
Research,
(Received 5 December 1994; in revised,form 20 June 1995)
Abstract-In
this paper we construct microstructures of multiphase composites with unusual properties : their heat conductivity in one direction is equal to the harmonic or arithmetic mean of the phases’ heat conductivities and the conductivity in an orthogonal direction does not equal either arithmetic or harmonic mean. Two-dimensional three-phase structures are studied, but the results can be easily generalized for the three-dimensional composites with arbitrary number of phases. Copyright 0 1996 Elsevier Science Ltd. I. INTRODUCTION
The paper is concerned with the structures of multicomponent composites with extremal effective properties. To explain what the “extremal” properties are we use the concept of the G,-closure set. The G,-closure is the set of the effective properties tensors of all mixtures (composites) that can be assembled from the given amounts of the given materials. For the heat conductivity problem, each effective conductivity tensor is determined by its eigenvalues (principal heat conductivities) (A,, A2,. . . ,), and the G,-closure is presented as a set of these vectors. The G,-closure set is definitely bounded ; its boundary corresponds to the composites with extremal properties. Now we call “extremal” the composite structures which correspond to the boundary of the G,-closure. For the two-dimensional heat conductivity problem, one can visualize the G, closure as a two-dimensional domain in the space of eigenvalues of effective heat conductivity tensors. The boundary points of G,-closure possess extremal overall properties. Therefore we call them “extremal points”, and corresponding structures are called “extremal structures”. The G,-closure problem has attracted a lot of attention in recent years. The most complete results have been obtained for two-phase composites. Namely, the problem for the heat conducting materials has been completely solved for the mixtures of two isotropic phases, which means that all structures with extremal properties have been determined. The isotropic component of the G,-closure has been obtained by Hashin and Shtrikman (1962) ; the full set of anisotropic composites has been found by Lurie and Cherkaev (1984b, 1986), and Tartar (1985). The set of all pairs of electrical and magnetic permeability tensors of two-dimensional two-phase composites has been determined by Cherkaev and Gibiansky (1992) [the isotropic component of this set has been described earlier by Milton (198 la,b)]. Here the coupling effect has been detected. Bounds on the complex conductivity of an isotropic two-dimensional composite have been obtained by Milton (198 1a). In elasticity, Hashin and Shtrikman have found the smallest rectangle in the bulkshear moduli plane that contains the isotropic component of the G,-closure for a twophase composite. Milton and Phan-Thien (1982) and Cherkaev and Gibiansky (1993) have described a smaller set inside of this rectangle which represents more restrictive coupled bounds on the effective elastic moduli. The viscoelasticity problem has been studied by Gibiansky and Milton (1993), etc. 2609
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A. V. Cherkaev
and L. V. Gibiansky
Most of the previously mentioned results have been obtained by using variational techniques (the Hashin-Shtrikman method, the Translation Method or their modifications) for constructing geometrically independent bounds for composite properties. These inequalities restrict the G,-closure from outside. The optimality of the bounds has been proved by consideration of the special classes of microstructures which allow explicit calculation of their effective properties. The set of the effective properties of those classes restricts the G,-closure from inside. When these two sets coincide, the complete description of the G,-closure is achieved. Relevant results for the multiphase composites are much weaker; there exists a gap between the already obtained bounds and the set of the properties of the known classes of microstructures. We mention here some papers where the reader can find results and further references: the bounds for the conductivity of isotropic multiphase mixtures have been found by Hashin and Shtrikman (1962), attainability of the Hashin-Shtrikman bounds has been studied by Milton (198 lc) and by Kohn and Milton (1988). Microstructures of optimal isotropic conductors have been studied by Lurie and Cherkaev (1985). Bounds on the effective properties of an anisotropic multiphase composite have been found by Zhikov (1986). The most complete analyses of the bounds and their attainability for the anisotropic mixtures can be found in the paper by Kohn and Milton (1988). The complex conductivity case has been studied by Golden and Papanicolaou (1985) and Milton and Golden (1990). Some bounds for the complex conductivity of multiphase composites have been given by Golden (1986) and by Milton (1987). Recent progress in the problem has been achieved by Nesi (1994). He has obtained a set of geometrically independent bounds on the conductivity of multiphase two-dimensional composites that improve upon the Hashin-Shtrikman results. The idea of his approach is close to the translation method. It is based on exploiting the special regularity properties of the solution of the conductivity equations in two dimensions that have been proved in the same paper. However, in spite of numerous efforts, even the simplest problem of the exact bounds on the effective conductivity of a two-dimensional composite made of three isotropic materials is still mainly open: the gap between the bounds and the set of the known structures still exists. In the present paper we consider one part of this problem: we explicitly construct anisotropic heat conducting structures (special laminates) which possess the extremal value of one of the principal heat conductivities. Therefore, these structures are certainly optimal : they belong to the boundary of the G,-closure. Suggested structures correspond to the boundary points ; the existence of these structures proves that the boundary component is exact. Also we formulate some rules of assembling extremal mixtures. The construction of the microstructures exploits the idea of an “imitation” that was suggested by Schulgasser ( 1976) and then used by Milton (198 lc, 1987), and by Lurie and Cherkaev (1985). For the sake of brevity we restrict ourselves in this paper to the particular case of a three-phase two-dimensional heat conducting composite. We focus our attention on composites with extremal value of one of the principal conductivities (they clearly form a component of the G,-closure boundary) and we compare extremal two- and three-phases mixtures. Two-phase mixtures in two dimensions possess the following property: if the heat conductivity of the composite in some direction is equal to the harmonic or the arithmetic mean, then the conductivity in an orthogonal direction is equal to the arithmetic or the harmonic mean, respectively. Surprisingly, this is not true for multicomponent mixtures. We construct the heat conducting multiphase composites of a special structure with the following properties : the conductivity in one direction is extreme and equal to the harmonic mean of the phases’ conductivities while the conductivity in an orthogonal direction is less than an arithmetic mean! Although we consider the simplest problem, the results obtained have more general significance. First of all, they can be immediately applied to the anti-plane strain problem which is formally identical to the heat conductivity problem in two dimensions (the equations of these problems coincide if the heat conductivities are replaced by the inverse shear moduli, and the temperature is replaced by the stress potential). Also, the plane elasticity
Extremal
structures
of multiphase
heat conducting
composites
2611
problem for materials with equal bulk moduli can be reduced to the problem discussed here, see Lurie and Cherkaev (1984a) for details. But even more important here seems to be a demonstration of principles of construction of optimal geometries for multicomponent composites. These principles are definitely not limited to the considered example but are applicable to any multicomponent mixture. The optimal structures that we have found have no analogues for two-component mixtures. The plan of the paper is the following: in Section 2 we give the statement of the problem, Section 3 summarizes known results, Section 4 contains the main results of the paper, namely, the description of extremal microstructures that form a component of the G,,,-closure boundary.
2. STATEMENT
OF THE
PROBLEM
Consider a three-phase heat conducting composite with periodic structure. Suppose that each cell of periodicity is divided into three parts R,, R,, and Q3 that have relative volume fractions m,, m,, and m3, m, fm, +m3 = 1. Suppose now that these parts are filled by three isotropic heat conducting materials with specific heat conductivities K,, K:, K+ Assume also that
The heat conductivity in this heterogeneous material is described by the Fourier law :
WI = We(x),
(2)
where if
XEQ,
if
XEQ*
if
XEQ~
(3)
q(x) and e(x) are the heat flux and temperature gradient, respectively, i.e. divq = 0, Homogenization
e(x) = VT.
(4)
of the heat conductivity in such a media leads to the relationship
(04)
= K*
(5)
where (.) denotes averaging over the periodic cell, K* is the effective heat conductivity tensor. This tensor K* depends on the conductivity constants K, of the phases, on their volume fractions mj in the mixture, and on the microstructure. The tensor K* (which is second order, possibly anisotropic, symmetric, and positive definite) is characterized by its eigenvalues 2, and & and the eigenvectors (these have no importance for us). It has the representation :
(6) in the basis associated with the eigenvectors of this tensor. The eigenvalues /2, and & are bounded by the arithmetic and harmonic means of the components conductivities, i.e. Kh
where
d 1, d
&,,
i=
1,2,
(7)
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Ich =
A. V. Cherkaev and L. V. Gibiansky
K,
=
rn,KI
+m,K,
+m,lc,.
(8)
These bounds are known and were proved by Wiener (19 12) [see also Christensen (1979)]. We call them arithmetic-harmonic mean (AHM) bounds because they bound the effective heat conductivity by the harmonic or arithmetic means of the components’ conductivities. The set of the pairs (I*,, A,) that satisfy the bounds (7) is represented by a square in the A,& plane. We call it AHM square. The AHM bounds are optimal in the following sense: they describe the minimal rectangle in the plane II,-/?, which contains the G,-closure set. To demonstrate this, consider a laminate structure which represents a point of the G,closure. One of the eigenvalues of the effective conductivity tensor of the laminates (the one that describes the conductivity in the direction n across the laminates) is equal to the harmonic mean of the components conductivities :
1 -I
IL, =
The other one, corresponding to the conductivity in the direction equal to the arithmetic mean of the components conductivities : 4
=
mlK1
(9)
t along the laminates,
-lrn,K,-Frn,K~.
is
(10)
This shows that two symmetric corners (IC,,,K,) and (K~, K,J of the AHM square correspond to the laminate structure and that the AHM square is the minimal one, indeed. Remark. The simple rules (9) and (10) for calculation of the effective conductivity of the laminates are the key (and the only one!) tool of our analyses. We call them harmonic and arithmetic mean rules, respectively. Note, that they are valid also for a laminate of anisotropic phases if the direction of lamination coincides with one of the principal axes of the phase’s conductivity tensors. There is a natural physical explanation of the laminate’s extremal properties. If one applies an external temperature gradient e along the layers then the local field e(x) in the laminates takes the same value in each point of the structure :
e(x) = const. (x) = e.
(11)
and q(x) =
Ic(x)e.
w-4
Therefore the effective conductivity along the laminates [which relates (e(x)) and (q(x))] is given by the arithmetic mean rule (as for the conductivity of conductors in parallel), i.e. (13)
(q(x)) == (K(X)h
Similarly, the external heat flux applied across the layers causes the constant local heat flux q(x) = const.(x) = q. Then the effective conductivity across the laminates is given by the harmonic mean rule (as for the conductivity of conductors in series) :
=
AHM
We are interesting square.
to find extremal
1 ~ t K(X) i q.
composites
that correspond
to the bounds
of the
Extremal
1.0 1
.u
structures
of multiphase
heat conducting
3.u
Rh
2613
composites
3.u
4
Effective heat conductivity, h, Fig.
1. G,-closure
set for the two-phase
two-dimensional problem. the AHM bounds.
3. KNOWN
Square
ABCD corresponds
to
BOUNDS
3.1. G,-closure for the two-phase composite
The G,-closure set for the two-component mixtures was found by Lurie and Cherkaev (1984b, 1986) and Tartar (1985). In particular, in two-dimensional space it is described by the following inequalities :
l+A+* 1.1-K2
(16) “2
2
and is illustrated in Fig. 1. [The previously mentioned Hashin and Shtrikman (1962) bounds on the effective conductivity of a two-phase isotropic composite coincide with the inequalities (15) and (16) restricted to the case A, = A,]. These bounds are optimal in a sense that they correspond to special microstructures, see Lurie and Cherkaev (1984b) and Tartar (1985). One can see that the G,-closure set is smaller than the AHM square. Only the corner point B = (JC,,,K,) and the symmetric point D = (K,, K~) correspond to structures. The only structure which belongs to the boundary of the AHM square is the laminate (we recall that the conductivity along the laminates is equal to K, and the conductivity across them is equal t0 Kh).
The slopes of the boundary curves (15) and (16) are strictly negative and finite. This implies, in particular, that the arithmetic mean rule for the effective conductivity in any direction implies the harmonic mean rule for the effective conductivity in the orthogonal direction and vice versa. This observation corresponds to the fact that an increase of the maximal possible conductivity of the extremal mixture in one direction necessarily leads to a decrease of the conductivity in the orthogonal direction and vice versa. It is surprising that the last conclusion is not true for the multiphase mixtures. In this work we present the microstructures that correspond to some other (not corner) points on the boundary of the AHM square for the three-phase mixture.
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A. V. Cherkaev and L. V. Gibiansky
3.2. Trace bounds for the multiphase
composite
Bounds on the effective conductivity of an isotropic multiphase composite have been found by Hashin and Shtrikman (1962). Namely, they found that the effective conductivity IC*of such a mixture is restricted by the inequalities KL d
K*
d
KU,
(17)
where (for the two-dimensional three-phase problem)
However, the attainability of the bounds for multicomponent structures is subject to some additional constraints. Namely, the lower bound (17) is exact and corresponds to some composite if xL < ICY,whereas the upper bound (17) is attainable if K” 3 x2, [see Milton (1981~) and Lurie and Cherkaev (1985)]. An anisotropic version of the Hashin-Shtrikman bounds has been obtained by Zhikov (1986) and Kohn and Milton (1988) who called them trace bounds. They have the form :
(20) The point (i1, &) on the lower bound (19) is known to be attainable if
where I,,,, = max {A,, A,} and E.,,, = min {A,, &} ; the point (A,, n,) on the upper bound (20) is attainable if
[see Kohn and Milton (1988)]. Figure 2 shows the A,-A2 plane where we draw the trace bounds (19) and (20) and mark the points (rcL, K”) and (I?, K~) of the Hashin-Shtrikman bounds for the following values of parameters : K,
=
1,
lc2 = 5,
ic3 = 25,
m, = 0.6,
m2 = 0.2,
m3 = 0.2.
(23)
The points on the bold part of the lower bound satisfy conditions (21) and, therefore, are attainable by some microstructures. At the same time one can check that conditions (22) (for the upper bound) are never satisfied for the chosen values of the parameters, therefore the attainability of this bound is not established. It should be noted, that bounds (19) and (20) are definitely not exact in the sense that they cannot be achieved for all values of the parameters IC,,K~, ic3, m,, m2, m3. For example, the lower bound does not satisfy natural limiting requirement
Gm(K1,rc2,K3,0,m2,m3) = G&,K3,m2,m3).
(24)
Here C?,(IC~,K~,ml, mJ is the set of the pairs (A,, A,) satisfying inequalities (15) and (16), and &,(G ICY,x3, ml, m2, m3) is the set of the pairs (,I,, 2,) that satisfy bounds (19) and
Extremal
structures
of multiphase
heat conducting
2615
composites
_F lcr=l, Ks=5, Ks=25
m,=O.6, mpO.2, mpO.2
1.0 1.0
3.0
5.0
7.0
Effective heat conductivity, h, Fig. 2. Bounds on the G,-closure set for the three-phase two-dimensional problem. The bold part of the lower bound EH is known to be attainable. In the scale of the figure, the points E,F,G,H (that mark the intersections of bounds (19) and (20) with the boundary of the AHM square ABCD) are indistinguishable from the points that correspond to the structures that we found.
(20). In other words, the established lower bound of the G,-closure does significantly depend on K,, even if the volume fraction of K, tends to zero. When the volume fraction reaches zero, the inequality has a discontinuity : this contradicts the expected continuous dependence of the bounds on the volume fractions.
Note that the new bounds recently suggested by Nesi (1994) satisfy equalities of the type (24). They are more restrictive than the Hashin-Shtrikman bounds although the attainability of these bounds is also not stated. The curves given by bounds (19) and (20) do not pass through the corner points of the AHM square (unlike the two-component case) ; instead they pass through the points on the sides of this square (points E, F, G, H in Fig. 2). It is important to emphasize that these bounds do not forbid the existence of the composites corresponding to the points on the sides of the AHM square. In the next section we demonstrate such structures. Remark.
4. COMPOSITE
STRUCTURES
WITH
EXTREMAL
PROPERTIES
In this section we find structures with one of their principal conductivities equal to the arithmetic mean bounds, 2, = K~,while the other one is bigger than the harmonic mean, /2, 3 K,,. These structures correspond to points on the intervals DG and BF on the side of the AHM square, see Fig. 2. Then by using similar arguments we find structures with A, = K,,,1, < K, which correspond to points on the intervals BE and DH, see Fig. 2. 4.1. Structures that support constant temperature gradient Let us begin with structures with principle conductivity in the direction x, equal to the arithmetic mean of the phase’s conductivities. These structures must support a constant temperature gradient when exposed to an external field directed along xl axis : it guarantees the equality ;1, = K,, see eqn (12). On the other hand, they should not be able to support a constant heat flux in the orthogonal direction x2. The last condition means that conductivity in the x2 direction should not be equal to the harmonic mean, i.e. j/* > ICY. We construct the structure that consists of three sequential laminations in the following order.
2616
A. V. Cherkaev
Fig. 3. Schematic
picture
and L. V. Gibiansky
of our three-step
process for creating
optimal
microstructures.
(1) First, we mix some amounts of the first and the third materials in proportions p and 1 -p in the laminates parallel to the axis x,, see Fig. 3(a). We choose the proportionp so that the conductivity of this laminate in the direction x, equals the conductivity of the intermediate material rcz,i.e.
Pi’) = plc1
+
(1 -p)rc, = K2.
(25)
Here and below, the lower index in the notation ;I!“’shows the direction, the upper index denotes the rank of the lamination. The required proportion p is equal to :
ton x2 across the laminates) is given by the The other eigenvalue ,I$‘) (in the dire c t’ harmonic mean rule and is equal to fly) =(p/lc, +(l-p)/Ic,)-1.
(27)
(2) At the second step we use the idea of imitation. We treat the previously obtained mixture as a new homogeneous material and mix it with the total amount available m2 of the material ICY in the laminates parallel to the axis x2, see Fig. 3(b). We assume that the scale of the second lamination is much larger than the scale of the lamination at the first step. Let us calculate the properties of the resulting composite. Both components have the same properties ICY and 25” = repin the direction x1. Therefore, the conductivity A\‘) in the direction x, (given by the harmonic mean rule) is equal to IQ, i.e.
2611
Extremal structures of multiphase heat conducting composites
q*
=
--I
-I
P
m2
(p+m,)A’,” + (P+m2)x2 1
= [ (p:lnz2, + (pZn2) 1
Ic2 =
K2.
(28)
Here ~1is the amount of the mixture of the first and third materials prepared at the first step of the process. We could say also that 2,(*) is equal to the arithmetic mean of the component’s conductivities A\” and K* (indeed, the arithmetic and the harmonic means of equal quantities trivially coincide!) : (29) The conductivity of the mixture in the other direction x2 is equal to the arithmetic mean of the component’s conductivities A$”and x2 and is given by the lamination formula :
(30) This step is the key point of the construction. Indeed, we obtain the composite with effective properties in all directions equal to the arithmetic mean of the properties of the component’s that enter the process at the second step. We have achieved this by a special choice of the intermediate material prepared at the first step. This material “imitates” the material K~ with respect to the conductivity in the direction xi and guarantees the arithmetic rule averaging (instead of the expected harmonic mean averaging) for the conductivity across the laminates. We follow here the idea exploited by Schulgasser (1976) for a threedimensional polycrystal structure. This idea has been used also by Milton (1981~) and by Lurie and Cherkaev (1985) for multiphase isotropic structures. They have considered isotropic mixtures that imitate the intermediate material K* with respect to the properties in all directions. Remark. Let us comment on the properties of the structure in terms of the applied fields. If one applies an external temperature gradient in the direction x, , the local field is constant throughout the composite because the properties of both components in this direction are equal to A\” = JC~,according to (28). Therefore, the conductivity 3L(12) of this structure in the direction x1 is equal to K*. On the other hand, if one applies the external temperature gradient in the direction x2, the heat flux is not constant and the conductivity of the mixture in that direction is different from the harmonic mean value rch.Indeed, the conductivities (in the direction x2) of the materials that are mixed at the second step differ, i.e. 1&”# K:. (3) To finish the construction let us laminate the already obtained amount v2 = p+m2
(31)
of the described composite with the remaining amounts h = ml --PP,
v3 = m3 -
(1 --PIP
(32)
of the first and the third materials, respectively. Now, we orient the lamination along the x, axis, see Fig. 3(c). Again, we assume that the scale of the third rank lamination is much larger than the scale of the lamination at the second step of the process. Applying the arithmetic and harmonic mean rules we find that the conductivity ,J\3)of this mixture in the direction x, is equal to nc,3'= V,K,+V2ni(,*'+VJK3.
One can substitute 2’1”=
ICYand
the values of the concentrations
(33) vi, v2, v3 into eqn (33) and
2618
A. V. Cherkaev and L. V. Gibiansky
check that the resulting conductivity ponents :
is given by the arithmetic mean of the initial com-
2’1” = m~lcl
fm2lc2 +m 3K3
(34)
[see eqns (29), (31) and (32)]. It is clear because this value is a result of three sequential arithmetic averagings. Physically, the external temperature gradient applied to the described composite along the x, axis causes the constant local field. This implies the equality (34). The other principal conductivity is equal to
43, =
VI + v2 p(p) [ Kl
+v’ K3
-1
I
= Ac,“@).
(35)
This value is equal to the harmonic mean ICY only if p = 0, otherwise it lies between Q, and K,. Therefore, we obtain the composite that corresponds to some point of the side of the AHM square but not to its corners. The effective conductivity of the obtained mixture depends on the amount p of the material involved at the first step of the process. More exactly, the eigenvalue A?’ = K, of the conductivity tensor is independent of p but the other eigenvalue A$”depends on it. One can check that @(p) monotonically increases as p increases. By changing /L one obtains an interval of the attainable points on the side of the AHM square. The value p = 0 corresponds to the simple laminate composite, i.e. to the corner points B = (q,, IC,) or D = (JC,, IQ,)of the AHM square. The maximal value pmaxcorresponds to the other end of the interval. The maximum amount p allowed by this construction is equal to
p,,=mink,
&}=min{rn,z,
rn3z),
[see eqn (32)]. Indeed, if
w/p6 m3/(l-14,
(37)
or, equivalently, m,(x,--K,)-m3(7c3--2)
GO,
(38)
then ,u is restricted by the available amount m, of the first phase. If, on the contrary, ml(~2-rc1)-m3(lc3--2)
20,
(39)
then the value p is restricted by the available amount m3 of the third phase. The structures corresponding to these two different limiting cases
(40)
Extremal
structures
of multiphase
heat conducting
composites
a>
2619
b)
Fig. 4. Schematic picture of the extremal microstructures corresponding H. Part a (b) corresponds to the higher (lower) volume fraction
to the points E, F, G, or of the third phase.
and PmaX= PU = %l(l -P)
(41)
are shown schematically in Figs 4(a, b), respectively. The maximum value of the effective conductivity A$31is equal to &3’(p’) i,,,
if eqn (38) holds,
= Ay)(pL,,,) =
A’,‘)($‘) ifeqn (39) holds.
(42)
In summary, we have found the composites corresponding to any point of the intervals
on the boundary of the AHM square. 4.2. Structures that support constant heatJux In this section we describe the composites with the principal heat conductivity in the direction x2 equal to the harmonic mean K,,: AZ = q,. Consider the same three-steps lamination procedure. At the first step we laminate the first and the third materials. We choose the fraction fi so that the conductivity &” across the layers is equal to rcz,i.e.
[compare with eqn (25)]. This proportion is equal to : B =(K;’ -K;‘)/(K:’
-XT’),
where@E[O, 1]
(45)
[compare with eqn (26)]. The other principal conductivity fi$” is given by the arithmetic mean rule p
Now at the second &‘I = ICY in the direction Again, the arithmetic and coincide. The conductivity mean of the component’s
=pK, +(I-_d)K3.
(46)
step both components possess the same properties K~ and x2. Therefore, the conductivity &*) in this direction equals K*. harmonic means of the equal quantities &I’ = rc2and ~~ trivially of the mixture in the other direction x, is equal to the harmonic conductivities I\” and ICYgiven by the lamination formula :
2620
A. V. Cherkaev
and L. V. Gibiansky m2
+ (P+m2)K2
1 I
(47)
Here @is the amount of the mixture of the first and the third materials prepared at the first step of the process. As we see, at the second step of this procedure we obtain a composite with effective properties in all directions equal to the harmonic mean of the conductivities of the components that enter the process at the second step. The intermediate material prepared at the first step “imitates” the material rc2with respect to the conductivity in the direction x2. To finish the construction we laminate the obtained amount C2= ,i+m, of the described composite with the remaining amounts v^,= m, -p/i, and C3= m3 - (1 -jS)fl of the first and the third materials, respectively. Applying the harmonic mean rule we obtain “(3)of this mixture in the direction x2 as the conductivity A2
(48)
The other principal conductivity is equal to
I\‘)@)= C,K, +v^242)+83K3.
(49)
This value is generally less than the arithmetic mean K, (they coincide only if ji = 0). Therefore, we have constructed a composite that corresponds to the point on the other side of the boundary of the AHM square. Again, the effective heat conductivity j(l’) depends on the amount & of the material prepared at the first step of the process. One can check, that i\3)(fi) monotonically decreases as fi increases. The value fi = 0 corresponds to the simple laminate composite, i.e. to the corner point B = (Q K,) or B = (K,, qJ of the AHM square. As in the previous section one can check that the maximum amount ,i2allowed by the available resources is equal to
[compare with eqn (36)]. The minimum value of the effective conductivity 2:” is equal to
/I,, = @(pma,J.
(51)
In summary, we have found the composites corresponding to any point of the intervals
on the sides of the AHM square.
5. DISCUSSION
The composites obtained may cover a significant part of the boundary of the AHM square. For example, consider the case when rc3is infinitely large (superconducting phase). Then rc, = cc and this square also becomes infinitely large :
Extremal
structures
of multiphase
heat conducting
2621
composites
The obtained structures cover semi-infinite intervals (52) on the sides of the AHM square if, in addition, &,,,, < co or, equivalently, mlK2
B(m, +m,)K,.
(54)
Indeed, one can check that in this case the extremal structure with the eigenvalues I., = K,, E., = /iminthat we have found possesses finite conductivities in both directions ; intervals (52) are semi-infinite. We calculate the heat conductivities of the extremal composites for the parameters given by eqn (23) and mark them on Fig. 2. In the scale of Fig. 2 the points corresponding to the optimal microstructures virtually coincide with the points E, F, G, and H of the trace bounds. It is not always so. Figure 5 illustrates the different situation when K,
=
1,
lc2 = 5,
rc3 = 25,
m, =
0.004,
m2 =
0.4,
m, =
0.596.
(55)
For these parameters the attainability conditions (22) are satisfied for the most part of the upper bound (bold line on Fig. 5), the microstructures that we found lie very close to the trace upper bound. The attainability conditions (21) failed in all the points of the lower bound. The distance between our structures (points Q and R) and the lower bound (points E and H) is very large. It is not surprising because the lower bound is expected to be crude for this choice of parameters [which comes close to realizing the condition of the test (24)]. From numerical experiments we have evidence that if the condition rcL6 JC~of attainability of the isotropic point of the Hashin-Shtrikman lower bound holds, our extremal structures lie close to the intersection of the lower bounds (19) with the boundary of the AHM square. The same is true for the upper bound (20) provided the inequality K” > K?
h
~t=l,
%=5,
tc3=25,
m,=O.04,
m&4,
m&596
B 2
s;
14.0
-
Q ‘S s T 8 iii 2 .-22 10.0 - E H El .3
‘\ \
\
\
\_ $y \\\,,\_ H
6.0
k 6.0
8.0
10.0
R 12.0
14.0
16.0X,
Effective heat conductivity, h, Fig. 5. Bounds on the G,-closure set for the three-phase two-dimensional problem. The bold part of the upper bound FG is known to be attainable. In the scale of the figure, the points F and G [that mark the intersections of the upper bound (20) with the boundary of the ANM square ABCD] are virtually indistinguishable from the points that correspond to the structures that we found. The points Q and R correspond to the extremal microstructures described in the text. Dashed line corresponds to the curve /2,/2, = I,&, where 1,, and 120 are the moduli of the point Q. This corresponds to a polycrystal comprised of grains of material Q.
2622
A. V. Cherkaev
and L. V. Gibiansky
holds. Figure 2 shows that our structures can lie close to the trace bound even when this condition is not met. We have described here some components of the boundary of the G,,-closure. The remaining components are still unknown. The G,-closure is, however, bounded from outside by inequalities (19) and (20) mentioned above. On the other hand, following an approach similar to that discussed here, one can consider other special laminate composites and, at least, bound the G,,-closure from inside. We will not guess here what the extremal structures are, but clearly the boundary of the set of pairs of their eigenvalues connects the obtained points, as schematically shown on Fig. 5 by the dashed line. This line represents the heat conductivity of the polycrystals made of the extremal composite corresponding to the point Q. [The effective conductivity of any two-dimensional polycrystal has the fixed value of the product of eigenvalues, see Lurie and Cherkaev (1981).] Note that these structures are used here because they are simple, not because they are extremal. The optimal structures for this component are still unknown. At this point we have not proved that there are no any microgeometries corresponding to the points in the intervals PQ and PM. This possibility cannot be ruled out based on our analyses. The obtained results can be easily generalized for the three-dimensional composite assembled of more than two phases. Indeed, the same construction (which exploits the idea of imitation) is directly applicable to the three-dimensional problem. Following the described scheme, one can obtain anisotropic structures which possess the harmonic mean conductivity in X, direction, the arithmetic mean conductivity in an orthogonal direction x1, and a conductivity in the third direction xj which is less than the arithmetic mean but belongs to the interval (52). These structures correspond to cylindrical geometries with the cross-section along (x,, x3) plane which is identical to the above described two-dimensional structures (Figs 1 and 2). Similarly, one can find structures which possess arithmetic mean heat conductivity in two orthogonal directions, but with the conductivity in the third direction which belongs to the interval (43). In other words, we are able to show the attainability of all points of the AHM cube in a neighborhood of the corner points (A,, /1,, &). (&,, i,,, j.,), and (j_,,, i.,, &). Ackno~~ledgements-Both authors gratefully acknowledge the support of the National Science Foundation under grant no. DM 593 07324. L. G. also gratefully acknowledges the support of the Air Force Office of Scientific Research under grant no. F49620-92-J-0501.
REFERENCES Christensen, R. M. (1979). Mechanics of Composite Materials. New York : Wiley-Interscience. Cherkaev, A. V. and Gibiansky, L. V. (1992). The exact coupled bounds for effective tensors of electrical and magnetic properties of two-component two-dimensional composites. Proc. R. Sec. Edinburgh 122A, 93-125. Cherkaev, A. V. and Gibiansky, L. V. (1993). Coupled estimates for the bulk and shear moduli of a twodimensional isotropic elastic composite. J. Mech. Phys. Solids 41(5), 937-980. Gibiansky, L. V. and Milton, G. W. (1993). On the effective viscoelastic moduli of two-phase media: I. Rigorous bounds on the complex bulk modulus. Proc. R. Sot. Lond. A440, 163-188. Golden, K. (1986). Bounds on the complex permittivity of a multicomponent material. J. Mech. Phys. Solids 34(4), 333-358. Golden, K. and Papanicolaou, G. (1985). Bounds for effective parameters of multicomponent media by analytic continuation. J. Stat. Phys. 40,655%667. Hashin, Z. and Shtrikman, S. (1962). A variational approach to the theory of the effective magnetic permeability of multiphase materials. J. Appl. Phys. 35, 3 125-3 13 1. Kohn. R. V. and Milton, G. W. (1988). Variational bounds on the effective moduli of anisotropic composites. J. Mech. Phys. Solids 36(6), 591-629. Lurie, K. A. and Cherkaev, A. V. (1981). G-closure of a set of anisotropically conducting media in the twodimensional case. Doklady Akademii Nauk SSSR 259(2), 238-241. Lurie, K. A. and Cherkaev, A. V. (1984a). G-closure of some particular sets of admissible material characteristics for the problem of bending of the thin elastic plates. J. Opt: Theory Appl. 42(2), 305-316. Lurie. K. A. and Cherkaev. A. V. (1984b). Exact estimates of conductivity of composites formed by two iso;ropically conducting media taken in pr&.cribed proportion. Proc. R. Soc.Edinburgi WA, 71-87. Lurie, K. A. and Cherkaev, A. V. (1985). The problem of formation of an optimal isotropic multicomponent composite. J. Opt. Theory Appl. 46, 571. Lurie, K. A. and Cherkaev, A. V. (1986). Exact estimates of the conductivity of a binary mixture of isotropic materials. Proc. R. Sot. Edinburgh 104A, 21-38.
Extremal
structures
of multiphase
heat conducting
composites
2623
Milton, G. W. (1981a). Bounds on the complex permittivity of two-component composite material. Appl. Phys. 52,528&5293. Milton, G. W. (1981b). Bounds on transport and optical properties of a two-component composite material. Appl. Phys. 52,5294-53&l. Milton, G. W. (198 1c). Concerning bounds on transport and mechanical properties of multicomponent composite materials. Appl. Phys. A26, 123. Milton. G. W. (1987). Multicomponent composites, impedance networks and new types of continued fraction I. II. Comm. Math. Phys. 111,281-327, 329-312. Milton, G. W. and Golden, K. (1990). Representation for the conductivity functions of multicomponent composites. Comm. Pure Appl. Math. l&647-671. Milton, G. W. and Phan-Thien, N. (1982). New bounds on the elastic moduli of two-component materials. Proc. R. Sot. Lond. A380,305-33 1. Nesi, V. (1994). Bounds on the effective conductivity of 2d composites made n > 3 isotropic phases in prescribed volume fraction : the weighted translation method. Report 65, Department of Pure and Applied Mathematics. University of L’Aquila. Schulgasser, K. (1976). Relationship between single-crystal and polycrystal electrical conductivity. Appl. Phys. 47, 1880-1886. Tartar, L. (1985). Estimations fines des coefficients homogeneises. In Ennio de Giorgi Colloquium (Edited by P. Kree), Pitman Research Notes in Mathematics, Vol. 125, pp. 1688187. Wiener, W. (1912). Abhandlungen der Mathematisch Physischen Klasse der Koniglichen Sachsischen Gesellschaft der Wissenschaften, 32, 509 pp. Zhikov. V. (1986). On the estimates for the trace of a homogenized matrix. Math. Notes 40, 628-634.
Pergamon 0020-7683(95)00176-X
EXTREMAL
STRUCTURES CONDUCTING
J. Solidr Strucrures
Vol. 33, No. 18, pp. 2609-2623, 1996 0 1996 Elsevier Science Ltd Copyright Pnnted-in &at Britain. All rights reserved 002&7683/96 $15.00 + .oO
OF MULTIPHASE COMPOSITES
HEAT
A. V. CHERKAEV University
of Utah, Department
of Mathematics,
Salt Lake City, UT 84112, U.S.A.
and L. V. GIBIANSKY Princetown
University,
Department of Civil Engineering Princeton, NJ 08544, U.S.A.
and Operations
Research,
(Received 5 December 1994; in revised,form 20 June 1995)
Abstract-In
this paper we construct microstructures of multiphase composites with unusual properties : their heat conductivity in one direction is equal to the harmonic or arithmetic mean of the phases’ heat conductivities and the conductivity in an orthogonal direction does not equal either arithmetic or harmonic mean. Two-dimensional three-phase structures are studied, but the results can be easily generalized for the three-dimensional composites with arbitrary number of phases. Copyright 0 1996 Elsevier Science Ltd. I. INTRODUCTION
The paper is concerned with the structures of multicomponent composites with extremal effective properties. To explain what the “extremal” properties are we use the concept of the G,-closure set. The G,-closure is the set of the effective properties tensors of all mixtures (composites) that can be assembled from the given amounts of the given materials. For the heat conductivity problem, each effective conductivity tensor is determined by its eigenvalues (principal heat conductivities) (A,, A2,. . . ,), and the G,-closure is presented as a set of these vectors. The G,-closure set is definitely bounded ; its boundary corresponds to the composites with extremal properties. Now we call “extremal” the composite structures which correspond to the boundary of the G,-closure. For the two-dimensional heat conductivity problem, one can visualize the G, closure as a two-dimensional domain in the space of eigenvalues of effective heat conductivity tensors. The boundary points of G,-closure possess extremal overall properties. Therefore we call them “extremal points”, and corresponding structures are called “extremal structures”. The G,-closure problem has attracted a lot of attention in recent years. The most complete results have been obtained for two-phase composites. Namely, the problem for the heat conducting materials has been completely solved for the mixtures of two isotropic phases, which means that all structures with extremal properties have been determined. The isotropic component of the G,-closure has been obtained by Hashin and Shtrikman (1962) ; the full set of anisotropic composites has been found by Lurie and Cherkaev (1984b, 1986), and Tartar (1985). The set of all pairs of electrical and magnetic permeability tensors of two-dimensional two-phase composites has been determined by Cherkaev and Gibiansky (1992) [the isotropic component of this set has been described earlier by Milton (198 la,b)]. Here the coupling effect has been detected. Bounds on the complex conductivity of an isotropic two-dimensional composite have been obtained by Milton (198 1a). In elasticity, Hashin and Shtrikman have found the smallest rectangle in the bulkshear moduli plane that contains the isotropic component of the G,-closure for a twophase composite. Milton and Phan-Thien (1982) and Cherkaev and Gibiansky (1993) have described a smaller set inside of this rectangle which represents more restrictive coupled bounds on the effective elastic moduli. The viscoelasticity problem has been studied by Gibiansky and Milton (1993), etc. 2609
2610
A. V. Cherkaev
and L. V. Gibiansky
Most of the previously mentioned results have been obtained by using variational techniques (the Hashin-Shtrikman method, the Translation Method or their modifications) for constructing geometrically independent bounds for composite properties. These inequalities restrict the G,-closure from outside. The optimality of the bounds has been proved by consideration of the special classes of microstructures which allow explicit calculation of their effective properties. The set of the effective properties of those classes restricts the G,-closure from inside. When these two sets coincide, the complete description of the G,-closure is achieved. Relevant results for the multiphase composites are much weaker; there exists a gap between the already obtained bounds and the set of the properties of the known classes of microstructures. We mention here some papers where the reader can find results and further references: the bounds for the conductivity of isotropic multiphase mixtures have been found by Hashin and Shtrikman (1962), attainability of the Hashin-Shtrikman bounds has been studied by Milton (198 lc) and by Kohn and Milton (1988). Microstructures of optimal isotropic conductors have been studied by Lurie and Cherkaev (1985). Bounds on the effective properties of an anisotropic multiphase composite have been found by Zhikov (1986). The most complete analyses of the bounds and their attainability for the anisotropic mixtures can be found in the paper by Kohn and Milton (1988). The complex conductivity case has been studied by Golden and Papanicolaou (1985) and Milton and Golden (1990). Some bounds for the complex conductivity of multiphase composites have been given by Golden (1986) and by Milton (1987). Recent progress in the problem has been achieved by Nesi (1994). He has obtained a set of geometrically independent bounds on the conductivity of multiphase two-dimensional composites that improve upon the Hashin-Shtrikman results. The idea of his approach is close to the translation method. It is based on exploiting the special regularity properties of the solution of the conductivity equations in two dimensions that have been proved in the same paper. However, in spite of numerous efforts, even the simplest problem of the exact bounds on the effective conductivity of a two-dimensional composite made of three isotropic materials is still mainly open: the gap between the bounds and the set of the known structures still exists. In the present paper we consider one part of this problem: we explicitly construct anisotropic heat conducting structures (special laminates) which possess the extremal value of one of the principal heat conductivities. Therefore, these structures are certainly optimal : they belong to the boundary of the G,-closure. Suggested structures correspond to the boundary points ; the existence of these structures proves that the boundary component is exact. Also we formulate some rules of assembling extremal mixtures. The construction of the microstructures exploits the idea of an “imitation” that was suggested by Schulgasser ( 1976) and then used by Milton (198 lc, 1987), and by Lurie and Cherkaev (1985). For the sake of brevity we restrict ourselves in this paper to the particular case of a three-phase two-dimensional heat conducting composite. We focus our attention on composites with extremal value of one of the principal conductivities (they clearly form a component of the G,-closure boundary) and we compare extremal two- and three-phases mixtures. Two-phase mixtures in two dimensions possess the following property: if the heat conductivity of the composite in some direction is equal to the harmonic or the arithmetic mean, then the conductivity in an orthogonal direction is equal to the arithmetic or the harmonic mean, respectively. Surprisingly, this is not true for multicomponent mixtures. We construct the heat conducting multiphase composites of a special structure with the following properties : the conductivity in one direction is extreme and equal to the harmonic mean of the phases’ conductivities while the conductivity in an orthogonal direction is less than an arithmetic mean! Although we consider the simplest problem, the results obtained have more general significance. First of all, they can be immediately applied to the anti-plane strain problem which is formally identical to the heat conductivity problem in two dimensions (the equations of these problems coincide if the heat conductivities are replaced by the inverse shear moduli, and the temperature is replaced by the stress potential). Also, the plane elasticity
Extremal
structures
of multiphase
heat conducting
composites
2611
problem for materials with equal bulk moduli can be reduced to the problem discussed here, see Lurie and Cherkaev (1984a) for details. But even more important here seems to be a demonstration of principles of construction of optimal geometries for multicomponent composites. These principles are definitely not limited to the considered example but are applicable to any multicomponent mixture. The optimal structures that we have found have no analogues for two-component mixtures. The plan of the paper is the following: in Section 2 we give the statement of the problem, Section 3 summarizes known results, Section 4 contains the main results of the paper, namely, the description of extremal microstructures that form a component of the G,,,-closure boundary.
2. STATEMENT
OF THE
PROBLEM
Consider a three-phase heat conducting composite with periodic structure. Suppose that each cell of periodicity is divided into three parts R,, R,, and Q3 that have relative volume fractions m,, m,, and m3, m, fm, +m3 = 1. Suppose now that these parts are filled by three isotropic heat conducting materials with specific heat conductivities K,, K:, K+ Assume also that
The heat conductivity in this heterogeneous material is described by the Fourier law :
WI = We(x),
(2)
where if
XEQ,
if
XEQ*
if
XEQ~
(3)
q(x) and e(x) are the heat flux and temperature gradient, respectively, i.e. divq = 0, Homogenization
e(x) = VT.
(4)
of the heat conductivity in such a media leads to the relationship
(04)
= K*
(5)
where (.) denotes averaging over the periodic cell, K* is the effective heat conductivity tensor. This tensor K* depends on the conductivity constants K, of the phases, on their volume fractions mj in the mixture, and on the microstructure. The tensor K* (which is second order, possibly anisotropic, symmetric, and positive definite) is characterized by its eigenvalues 2, and & and the eigenvectors (these have no importance for us). It has the representation :
(6) in the basis associated with the eigenvectors of this tensor. The eigenvalues /2, and & are bounded by the arithmetic and harmonic means of the components conductivities, i.e. Kh
where
d 1, d
&,,
i=
1,2,
(7)
2612
Ich =
A. V. Cherkaev and L. V. Gibiansky
K,
=
rn,KI
+m,K,
+m,lc,.
(8)
These bounds are known and were proved by Wiener (19 12) [see also Christensen (1979)]. We call them arithmetic-harmonic mean (AHM) bounds because they bound the effective heat conductivity by the harmonic or arithmetic means of the components’ conductivities. The set of the pairs (I*,, A,) that satisfy the bounds (7) is represented by a square in the A,& plane. We call it AHM square. The AHM bounds are optimal in the following sense: they describe the minimal rectangle in the plane II,-/?, which contains the G,-closure set. To demonstrate this, consider a laminate structure which represents a point of the G,closure. One of the eigenvalues of the effective conductivity tensor of the laminates (the one that describes the conductivity in the direction n across the laminates) is equal to the harmonic mean of the components conductivities :
1 -I
IL, =
The other one, corresponding to the conductivity in the direction equal to the arithmetic mean of the components conductivities : 4
=
mlK1
(9)
t along the laminates,
-lrn,K,-Frn,K~.
is
(10)
This shows that two symmetric corners (IC,,,K,) and (K~, K,J of the AHM square correspond to the laminate structure and that the AHM square is the minimal one, indeed. Remark. The simple rules (9) and (10) for calculation of the effective conductivity of the laminates are the key (and the only one!) tool of our analyses. We call them harmonic and arithmetic mean rules, respectively. Note, that they are valid also for a laminate of anisotropic phases if the direction of lamination coincides with one of the principal axes of the phase’s conductivity tensors. There is a natural physical explanation of the laminate’s extremal properties. If one applies an external temperature gradient e along the layers then the local field e(x) in the laminates takes the same value in each point of the structure :
e(x) = const. (x) = e.
(11)
and q(x) =
Ic(x)e.
w-4
Therefore the effective conductivity along the laminates [which relates (e(x)) and (q(x))] is given by the arithmetic mean rule (as for the conductivity of conductors in parallel), i.e. (13)
(q(x)) =
Similarly, the external heat flux applied across the layers causes the constant local heat flux q(x) = const.(x) = q. Then the effective conductivity across the laminates is given by the harmonic mean rule (as for the conductivity of conductors in series) :
AHM
We are interesting square.
to find extremal
1 ~ t K(X) i q.
composites
that correspond
to the bounds
of the
Extremal
1.0 1
.u
structures
of multiphase
heat conducting
3.u
Rh
2613
composites
3.u
4
Effective heat conductivity, h, Fig.
1. G,-closure
set for the two-phase
two-dimensional problem. the AHM bounds.
3. KNOWN
Square
ABCD corresponds
to
BOUNDS
3.1. G,-closure for the two-phase composite
The G,-closure set for the two-component mixtures was found by Lurie and Cherkaev (1984b, 1986) and Tartar (1985). In particular, in two-dimensional space it is described by the following inequalities :
l+A+* 1.1-K2
(16) “2
2
and is illustrated in Fig. 1. [The previously mentioned Hashin and Shtrikman (1962) bounds on the effective conductivity of a two-phase isotropic composite coincide with the inequalities (15) and (16) restricted to the case A, = A,]. These bounds are optimal in a sense that they correspond to special microstructures, see Lurie and Cherkaev (1984b) and Tartar (1985). One can see that the G,-closure set is smaller than the AHM square. Only the corner point B = (JC,,,K,) and the symmetric point D = (K,, K~) correspond to structures. The only structure which belongs to the boundary of the AHM square is the laminate (we recall that the conductivity along the laminates is equal to K, and the conductivity across them is equal t0 Kh).
The slopes of the boundary curves (15) and (16) are strictly negative and finite. This implies, in particular, that the arithmetic mean rule for the effective conductivity in any direction implies the harmonic mean rule for the effective conductivity in the orthogonal direction and vice versa. This observation corresponds to the fact that an increase of the maximal possible conductivity of the extremal mixture in one direction necessarily leads to a decrease of the conductivity in the orthogonal direction and vice versa. It is surprising that the last conclusion is not true for the multiphase mixtures. In this work we present the microstructures that correspond to some other (not corner) points on the boundary of the AHM square for the three-phase mixture.
2614
A. V. Cherkaev and L. V. Gibiansky
3.2. Trace bounds for the multiphase
composite
Bounds on the effective conductivity of an isotropic multiphase composite have been found by Hashin and Shtrikman (1962). Namely, they found that the effective conductivity IC*of such a mixture is restricted by the inequalities KL d
K*
d
KU,
(17)
where (for the two-dimensional three-phase problem)
However, the attainability of the bounds for multicomponent structures is subject to some additional constraints. Namely, the lower bound (17) is exact and corresponds to some composite if xL < ICY,whereas the upper bound (17) is attainable if K” 3 x2, [see Milton (1981~) and Lurie and Cherkaev (1985)]. An anisotropic version of the Hashin-Shtrikman bounds has been obtained by Zhikov (1986) and Kohn and Milton (1988) who called them trace bounds. They have the form :
(20) The point (i1, &) on the lower bound (19) is known to be attainable if
where I,,,, = max {A,, A,} and E.,,, = min {A,, &} ; the point (A,, n,) on the upper bound (20) is attainable if
[see Kohn and Milton (1988)]. Figure 2 shows the A,-A2 plane where we draw the trace bounds (19) and (20) and mark the points (rcL, K”) and (I?, K~) of the Hashin-Shtrikman bounds for the following values of parameters : K,
=
1,
lc2 = 5,
ic3 = 25,
m, = 0.6,
m2 = 0.2,
m3 = 0.2.
(23)
The points on the bold part of the lower bound satisfy conditions (21) and, therefore, are attainable by some microstructures. At the same time one can check that conditions (22) (for the upper bound) are never satisfied for the chosen values of the parameters, therefore the attainability of this bound is not established. It should be noted, that bounds (19) and (20) are definitely not exact in the sense that they cannot be achieved for all values of the parameters IC,,K~, ic3, m,, m2, m3. For example, the lower bound does not satisfy natural limiting requirement
Gm(K1,rc2,K3,0,m2,m3) = G&,K3,m2,m3).
(24)
Here C?,(IC~,K~,ml, mJ is the set of the pairs (A,, A,) satisfying inequalities (15) and (16), and &,(G ICY,x3, ml, m2, m3) is the set of the pairs (,I,, 2,) that satisfy bounds (19) and
Extremal
structures
of multiphase
heat conducting
2615
composites
_F lcr=l, Ks=5, Ks=25
m,=O.6, mpO.2, mpO.2
1.0 1.0
3.0
5.0
7.0
Effective heat conductivity, h, Fig. 2. Bounds on the G,-closure set for the three-phase two-dimensional problem. The bold part of the lower bound EH is known to be attainable. In the scale of the figure, the points E,F,G,H (that mark the intersections of bounds (19) and (20) with the boundary of the AHM square ABCD) are indistinguishable from the points that correspond to the structures that we found.
(20). In other words, the established lower bound of the G,-closure does significantly depend on K,, even if the volume fraction of K, tends to zero. When the volume fraction reaches zero, the inequality has a discontinuity : this contradicts the expected continuous dependence of the bounds on the volume fractions.
Note that the new bounds recently suggested by Nesi (1994) satisfy equalities of the type (24). They are more restrictive than the Hashin-Shtrikman bounds although the attainability of these bounds is also not stated. The curves given by bounds (19) and (20) do not pass through the corner points of the AHM square (unlike the two-component case) ; instead they pass through the points on the sides of this square (points E, F, G, H in Fig. 2). It is important to emphasize that these bounds do not forbid the existence of the composites corresponding to the points on the sides of the AHM square. In the next section we demonstrate such structures. Remark.
4. COMPOSITE
STRUCTURES
WITH
EXTREMAL
PROPERTIES
In this section we find structures with one of their principal conductivities equal to the arithmetic mean bounds, 2, = K~,while the other one is bigger than the harmonic mean, /2, 3 K,,. These structures correspond to points on the intervals DG and BF on the side of the AHM square, see Fig. 2. Then by using similar arguments we find structures with A, = K,,,1, < K, which correspond to points on the intervals BE and DH, see Fig. 2. 4.1. Structures that support constant temperature gradient Let us begin with structures with principle conductivity in the direction x, equal to the arithmetic mean of the phase’s conductivities. These structures must support a constant temperature gradient when exposed to an external field directed along xl axis : it guarantees the equality ;1, = K,, see eqn (12). On the other hand, they should not be able to support a constant heat flux in the orthogonal direction x2. The last condition means that conductivity in the x2 direction should not be equal to the harmonic mean, i.e. j/* > ICY. We construct the structure that consists of three sequential laminations in the following order.
2616
A. V. Cherkaev
Fig. 3. Schematic
picture
and L. V. Gibiansky
of our three-step
process for creating
optimal
microstructures.
(1) First, we mix some amounts of the first and the third materials in proportions p and 1 -p in the laminates parallel to the axis x,, see Fig. 3(a). We choose the proportionp so that the conductivity of this laminate in the direction x, equals the conductivity of the intermediate material rcz,i.e.
Pi’) = plc1
+
(1 -p)rc, = K2.
(25)
Here and below, the lower index in the notation ;I!“’shows the direction, the upper index denotes the rank of the lamination. The required proportion p is equal to :
ton x2 across the laminates) is given by the The other eigenvalue ,I$‘) (in the dire c t’ harmonic mean rule and is equal to fly) =(p/lc, +(l-p)/Ic,)-1.
(27)
(2) At the second step we use the idea of imitation. We treat the previously obtained mixture as a new homogeneous material and mix it with the total amount available m2 of the material ICY in the laminates parallel to the axis x2, see Fig. 3(b). We assume that the scale of the second lamination is much larger than the scale of the lamination at the first step. Let us calculate the properties of the resulting composite. Both components have the same properties ICY and 25” = repin the direction x1. Therefore, the conductivity A\‘) in the direction x, (given by the harmonic mean rule) is equal to IQ, i.e.
2611
Extremal structures of multiphase heat conducting composites
q*
=
--I
-I
P
m2
(p+m,)A’,” + (P+m2)x2 1
= [ (p:lnz2, + (pZn2) 1
Ic2 =
K2.
(28)
Here ~1is the amount of the mixture of the first and third materials prepared at the first step of the process. We could say also that 2,(*) is equal to the arithmetic mean of the component’s conductivities A\” and K* (indeed, the arithmetic and the harmonic means of equal quantities trivially coincide!) : (29) The conductivity of the mixture in the other direction x2 is equal to the arithmetic mean of the component’s conductivities A$”and x2 and is given by the lamination formula :
(30) This step is the key point of the construction. Indeed, we obtain the composite with effective properties in all directions equal to the arithmetic mean of the properties of the component’s that enter the process at the second step. We have achieved this by a special choice of the intermediate material prepared at the first step. This material “imitates” the material K~ with respect to the conductivity in the direction xi and guarantees the arithmetic rule averaging (instead of the expected harmonic mean averaging) for the conductivity across the laminates. We follow here the idea exploited by Schulgasser (1976) for a threedimensional polycrystal structure. This idea has been used also by Milton (1981~) and by Lurie and Cherkaev (1985) for multiphase isotropic structures. They have considered isotropic mixtures that imitate the intermediate material K* with respect to the properties in all directions. Remark. Let us comment on the properties of the structure in terms of the applied fields. If one applies an external temperature gradient in the direction x, , the local field is constant throughout the composite because the properties of both components in this direction are equal to A\” = JC~,according to (28). Therefore, the conductivity 3L(12) of this structure in the direction x1 is equal to K*. On the other hand, if one applies the external temperature gradient in the direction x2, the heat flux is not constant and the conductivity of the mixture in that direction is different from the harmonic mean value rch.Indeed, the conductivities (in the direction x2) of the materials that are mixed at the second step differ, i.e. 1&”# K:. (3) To finish the construction let us laminate the already obtained amount v2 = p+m2
(31)
of the described composite with the remaining amounts h = ml --PP,
v3 = m3 -
(1 --PIP
(32)
of the first and the third materials, respectively. Now, we orient the lamination along the x, axis, see Fig. 3(c). Again, we assume that the scale of the third rank lamination is much larger than the scale of the lamination at the second step of the process. Applying the arithmetic and harmonic mean rules we find that the conductivity ,J\3)of this mixture in the direction x, is equal to nc,3'= V,K,+V2ni(,*'+VJK3.
One can substitute 2’1”=
ICYand
the values of the concentrations
(33) vi, v2, v3 into eqn (33) and
2618
A. V. Cherkaev and L. V. Gibiansky
check that the resulting conductivity ponents :
is given by the arithmetic mean of the initial com-
2’1” = m~lcl
fm2lc2 +m 3K3
(34)
[see eqns (29), (31) and (32)]. It is clear because this value is a result of three sequential arithmetic averagings. Physically, the external temperature gradient applied to the described composite along the x, axis causes the constant local field. This implies the equality (34). The other principal conductivity is equal to
43, =
VI + v2 p(p) [ Kl
+v’ K3
-1
I
= Ac,“@).
(35)
This value is equal to the harmonic mean ICY only if p = 0, otherwise it lies between Q, and K,. Therefore, we obtain the composite that corresponds to some point of the side of the AHM square but not to its corners. The effective conductivity of the obtained mixture depends on the amount p of the material involved at the first step of the process. More exactly, the eigenvalue A?’ = K, of the conductivity tensor is independent of p but the other eigenvalue A$”depends on it. One can check that @(p) monotonically increases as p increases. By changing /L one obtains an interval of the attainable points on the side of the AHM square. The value p = 0 corresponds to the simple laminate composite, i.e. to the corner points B = (q,, IC,) or D = (JC,, IQ,)of the AHM square. The maximal value pmaxcorresponds to the other end of the interval. The maximum amount p allowed by this construction is equal to
p,,=mink,
&}=min{rn,z,
rn3z),
[see eqn (32)]. Indeed, if
w/p6 m3/(l-14,
(37)
or, equivalently, m,(x,--K,)-m3(7c3--2)
GO,
(38)
then ,u is restricted by the available amount m, of the first phase. If, on the contrary, ml(~2-rc1)-m3(lc3--2)
20,
(39)
then the value p is restricted by the available amount m3 of the third phase. The structures corresponding to these two different limiting cases
(40)
Extremal
structures
of multiphase
heat conducting
composites
a>
2619
b)
Fig. 4. Schematic picture of the extremal microstructures corresponding H. Part a (b) corresponds to the higher (lower) volume fraction
to the points E, F, G, or of the third phase.
and PmaX= PU = %l(l -P)
(41)
are shown schematically in Figs 4(a, b), respectively. The maximum value of the effective conductivity A$31is equal to &3’(p’) i,,,
if eqn (38) holds,
= Ay)(pL,,,) =
A’,‘)($‘) ifeqn (39) holds.
(42)
In summary, we have found the composites corresponding to any point of the intervals
on the boundary of the AHM square. 4.2. Structures that support constant heatJux In this section we describe the composites with the principal heat conductivity in the direction x2 equal to the harmonic mean K,,: AZ = q,. Consider the same three-steps lamination procedure. At the first step we laminate the first and the third materials. We choose the fraction fi so that the conductivity &” across the layers is equal to rcz,i.e.
[compare with eqn (25)]. This proportion is equal to : B =(K;’ -K;‘)/(K:’
-XT’),
where@E[O, 1]
(45)
[compare with eqn (26)]. The other principal conductivity fi$” is given by the arithmetic mean rule p
Now at the second &‘I = ICY in the direction Again, the arithmetic and coincide. The conductivity mean of the component’s
=pK, +(I-_d)K3.
(46)
step both components possess the same properties K~ and x2. Therefore, the conductivity &*) in this direction equals K*. harmonic means of the equal quantities &I’ = rc2and ~~ trivially of the mixture in the other direction x, is equal to the harmonic conductivities I\” and ICYgiven by the lamination formula :
2620
A. V. Cherkaev
and L. V. Gibiansky m2
+ (P+m2)K2
1 I
(47)
Here @is the amount of the mixture of the first and the third materials prepared at the first step of the process. As we see, at the second step of this procedure we obtain a composite with effective properties in all directions equal to the harmonic mean of the conductivities of the components that enter the process at the second step. The intermediate material prepared at the first step “imitates” the material rc2with respect to the conductivity in the direction x2. To finish the construction we laminate the obtained amount C2= ,i+m, of the described composite with the remaining amounts v^,= m, -p/i, and C3= m3 - (1 -jS)fl of the first and the third materials, respectively. Applying the harmonic mean rule we obtain “(3)of this mixture in the direction x2 as the conductivity A2
(48)
The other principal conductivity is equal to
I\‘)@)= C,K, +v^242)+83K3.
(49)
This value is generally less than the arithmetic mean K, (they coincide only if ji = 0). Therefore, we have constructed a composite that corresponds to the point on the other side of the boundary of the AHM square. Again, the effective heat conductivity j(l’) depends on the amount & of the material prepared at the first step of the process. One can check, that i\3)(fi) monotonically decreases as fi increases. The value fi = 0 corresponds to the simple laminate composite, i.e. to the corner point B = (Q K,) or B = (K,, qJ of the AHM square. As in the previous section one can check that the maximum amount ,i2allowed by the available resources is equal to
[compare with eqn (36)]. The minimum value of the effective conductivity 2:” is equal to
/I,, = @(pma,J.
(51)
In summary, we have found the composites corresponding to any point of the intervals
on the sides of the AHM square.
5. DISCUSSION
The composites obtained may cover a significant part of the boundary of the AHM square. For example, consider the case when rc3is infinitely large (superconducting phase). Then rc, = cc and this square also becomes infinitely large :
Extremal
structures
of multiphase
heat conducting
2621
composites
The obtained structures cover semi-infinite intervals (52) on the sides of the AHM square if, in addition, &,,,, < co or, equivalently, mlK2
B(m, +m,)K,.
(54)
Indeed, one can check that in this case the extremal structure with the eigenvalues I., = K,, E., = /iminthat we have found possesses finite conductivities in both directions ; intervals (52) are semi-infinite. We calculate the heat conductivities of the extremal composites for the parameters given by eqn (23) and mark them on Fig. 2. In the scale of Fig. 2 the points corresponding to the optimal microstructures virtually coincide with the points E, F, G, and H of the trace bounds. It is not always so. Figure 5 illustrates the different situation when K,
=
1,
lc2 = 5,
rc3 = 25,
m, =
0.004,
m2 =
0.4,
m, =
0.596.
(55)
For these parameters the attainability conditions (22) are satisfied for the most part of the upper bound (bold line on Fig. 5), the microstructures that we found lie very close to the trace upper bound. The attainability conditions (21) failed in all the points of the lower bound. The distance between our structures (points Q and R) and the lower bound (points E and H) is very large. It is not surprising because the lower bound is expected to be crude for this choice of parameters [which comes close to realizing the condition of the test (24)]. From numerical experiments we have evidence that if the condition rcL6 JC~of attainability of the isotropic point of the Hashin-Shtrikman lower bound holds, our extremal structures lie close to the intersection of the lower bounds (19) with the boundary of the AHM square. The same is true for the upper bound (20) provided the inequality K” > K?
h
~t=l,
%=5,
tc3=25,
m,=O.04,
m&4,
m&596
B 2
s;
14.0
-
Q ‘S s T 8 iii 2 .-22 10.0 - E H El .3
‘\ \
\
\
\_ $y \\\,,\_ H
6.0
k 6.0
8.0
10.0
R 12.0
14.0
16.0X,
Effective heat conductivity, h, Fig. 5. Bounds on the G,-closure set for the three-phase two-dimensional problem. The bold part of the upper bound FG is known to be attainable. In the scale of the figure, the points F and G [that mark the intersections of the upper bound (20) with the boundary of the ANM square ABCD] are virtually indistinguishable from the points that correspond to the structures that we found. The points Q and R correspond to the extremal microstructures described in the text. Dashed line corresponds to the curve /2,/2, = I,&, where 1,, and 120 are the moduli of the point Q. This corresponds to a polycrystal comprised of grains of material Q.
2622
A. V. Cherkaev
and L. V. Gibiansky
holds. Figure 2 shows that our structures can lie close to the trace bound even when this condition is not met. We have described here some components of the boundary of the G,,-closure. The remaining components are still unknown. The G,-closure is, however, bounded from outside by inequalities (19) and (20) mentioned above. On the other hand, following an approach similar to that discussed here, one can consider other special laminate composites and, at least, bound the G,,-closure from inside. We will not guess here what the extremal structures are, but clearly the boundary of the set of pairs of their eigenvalues connects the obtained points, as schematically shown on Fig. 5 by the dashed line. This line represents the heat conductivity of the polycrystals made of the extremal composite corresponding to the point Q. [The effective conductivity of any two-dimensional polycrystal has the fixed value of the product of eigenvalues, see Lurie and Cherkaev (1981).] Note that these structures are used here because they are simple, not because they are extremal. The optimal structures for this component are still unknown. At this point we have not proved that there are no any microgeometries corresponding to the points in the intervals PQ and PM. This possibility cannot be ruled out based on our analyses. The obtained results can be easily generalized for the three-dimensional composite assembled of more than two phases. Indeed, the same construction (which exploits the idea of imitation) is directly applicable to the three-dimensional problem. Following the described scheme, one can obtain anisotropic structures which possess the harmonic mean conductivity in X, direction, the arithmetic mean conductivity in an orthogonal direction x1, and a conductivity in the third direction xj which is less than the arithmetic mean but belongs to the interval (52). These structures correspond to cylindrical geometries with the cross-section along (x,, x3) plane which is identical to the above described two-dimensional structures (Figs 1 and 2). Similarly, one can find structures which possess arithmetic mean heat conductivity in two orthogonal directions, but with the conductivity in the third direction which belongs to the interval (43). In other words, we are able to show the attainability of all points of the AHM cube in a neighborhood of the corner points (A,, /1,, &). (&,, i,,, j.,), and (j_,,, i.,, &). Ackno~~ledgements-Both authors gratefully acknowledge the support of the National Science Foundation under grant no. DM 593 07324. L. G. also gratefully acknowledges the support of the Air Force Office of Scientific Research under grant no. F49620-92-J-0501.
REFERENCES Christensen, R. M. (1979). Mechanics of Composite Materials. New York : Wiley-Interscience. Cherkaev, A. V. and Gibiansky, L. V. (1992). The exact coupled bounds for effective tensors of electrical and magnetic properties of two-component two-dimensional composites. Proc. R. Sec. Edinburgh 122A, 93-125. Cherkaev, A. V. and Gibiansky, L. V. (1993). Coupled estimates for the bulk and shear moduli of a twodimensional isotropic elastic composite. J. Mech. Phys. Solids 41(5), 937-980. Gibiansky, L. V. and Milton, G. W. (1993). On the effective viscoelastic moduli of two-phase media: I. Rigorous bounds on the complex bulk modulus. Proc. R. Sot. Lond. A440, 163-188. Golden, K. (1986). Bounds on the complex permittivity of a multicomponent material. J. Mech. Phys. Solids 34(4), 333-358. Golden, K. and Papanicolaou, G. (1985). Bounds for effective parameters of multicomponent media by analytic continuation. J. Stat. Phys. 40,655%667. Hashin, Z. and Shtrikman, S. (1962). A variational approach to the theory of the effective magnetic permeability of multiphase materials. J. Appl. Phys. 35, 3 125-3 13 1. Kohn. R. V. and Milton, G. W. (1988). Variational bounds on the effective moduli of anisotropic composites. J. Mech. Phys. Solids 36(6), 591-629. Lurie, K. A. and Cherkaev, A. V. (1981). G-closure of a set of anisotropically conducting media in the twodimensional case. Doklady Akademii Nauk SSSR 259(2), 238-241. Lurie, K. A. and Cherkaev, A. V. (1984a). G-closure of some particular sets of admissible material characteristics for the problem of bending of the thin elastic plates. J. Opt: Theory Appl. 42(2), 305-316. Lurie. K. A. and Cherkaev. A. V. (1984b). Exact estimates of conductivity of composites formed by two iso;ropically conducting media taken in pr&.cribed proportion. Proc. R. Soc.Edinburgi WA, 71-87. Lurie, K. A. and Cherkaev, A. V. (1985). The problem of formation of an optimal isotropic multicomponent composite. J. Opt. Theory Appl. 46, 571. Lurie, K. A. and Cherkaev, A. V. (1986). Exact estimates of the conductivity of a binary mixture of isotropic materials. Proc. R. Sot. Edinburgh 104A, 21-38.
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structures
of multiphase
heat conducting
composites
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Milton, G. W. (1981a). Bounds on the complex permittivity of two-component composite material. Appl. Phys. 52,528&5293. Milton, G. W. (1981b). Bounds on transport and optical properties of a two-component composite material. Appl. Phys. 52,5294-53&l. Milton, G. W. (198 1c). Concerning bounds on transport and mechanical properties of multicomponent composite materials. Appl. Phys. A26, 123. Milton. G. W. (1987). Multicomponent composites, impedance networks and new types of continued fraction I. II. Comm. Math. Phys. 111,281-327, 329-312. Milton, G. W. and Golden, K. (1990). Representation for the conductivity functions of multicomponent composites. Comm. Pure Appl. Math. l&647-671. Milton, G. W. and Phan-Thien, N. (1982). New bounds on the elastic moduli of two-component materials. Proc. R. Sot. Lond. A380,305-33 1. Nesi, V. (1994). Bounds on the effective conductivity of 2d composites made n > 3 isotropic phases in prescribed volume fraction : the weighted translation method. Report 65, Department of Pure and Applied Mathematics. University of L’Aquila. Schulgasser, K. (1976). Relationship between single-crystal and polycrystal electrical conductivity. Appl. Phys. 47, 1880-1886. Tartar, L. (1985). Estimations fines des coefficients homogeneises. In Ennio de Giorgi Colloquium (Edited by P. Kree), Pitman Research Notes in Mathematics, Vol. 125, pp. 1688187. Wiener, W. (1912). Abhandlungen der Mathematisch Physischen Klasse der Koniglichen Sachsischen Gesellschaft der Wissenschaften, 32, 509 pp. Zhikov. V. (1986). On the estimates for the trace of a homogenized matrix. Math. Notes 40, 628-634.